In general, in Android you can set things like notification tones and vibration patterns within the settings for each app.Within each app, tap the menu button, go into settings, and look for options for notifications.

I haven't used the default text message app in a long time, but I believe you can set a notification tone there. You can go one step further, and in the contacts app set a specific tone for individual contacts, too.

In the gmail app, go into settings, tap the account you want to change notifications for, and tap "Inbox sound & vibrate." In the Samsung email app, tap settings, the account, and then check Email notifications.

I have a Samsung Galaxy S3 and this issue has plagued me for the month & a half since I switched from my broken Iphone.
I BELIEVE I have fixed the problem. I eliminated all email accounts except for Gmail, including the default account (in my case, Comcast) Under the Gmail tab, General Settings, there is no selection for sound. So, I went to the main Settings menu / Sounds: and changed the default notification sound to Silent. Then in the default messaging Settings menu, under Notifications, I chose a sound for text messages. I do hope this works. I certainly don't want to be alerted every time I get an email or an app needs updating.

I am so far, extremely UNIMPRESSED with this phone. There is too much redundancy and overall, it's a clunker compared to the clean interface of the Iphone.

Someone smart should create an App that manages your notification sounds and allows you to set a different sound for things such as twitter, email, SMS and other crap you use on your S4.
They would make lots of bucks, i would be the first to use it.